The trend in modern machine manufacturing is to achieve ever higher capacity and output of the machines, which generally calls for higher rotational speeds and an optimum utilization of materials in the machines. As a result, dynamic problems in the operation of the machines, which have previously only been considered of secondary importance, have now come into the foreground. For these reasons, it is now especially important to consider vibration and oscillation processes, such as imbalance vibrations and flexural bending vibrations, in the construction of shafts, rolls and other rotors that are intended to operate at high rotational speeds.
While mass imbalances have been addressed in the prior art, the occurrence of flexural bending vibrations due to anisotropy in the flexural stiffness characteristic of a rotating member have not been adequately taken into account. Such flexural bending vibrations are especially critical in shafts or other rotating members that are to be operated at a speed approaching the first critical bending speed or especially more than half of that critical speed. An example is represented by the calendar rolls and other rolls used in paper manufacturing, which are characterized as elastic shafts, i.e. the roll or shaft cannot be regarded as rigid but rather is elastically flexible over its length. If such a shaft or roll has even a slight anisotropy in its flexural stiffness, it will undergo a very strong vibration having a frequency of twice (2f) the rotational frequency, once the rotational speed of the shaft or roll reaches about one half of the first critical bending speed. In the field of rotor dynamics, this phenomenon is known as a so-called weight critical resonance or weight critical behavior.
Since this phenomenon is caused by the anisotropic flexural stiffness characteristic, it can occur even if the shaft or roll appears to be perfectly "round" and properly mass balanced. In other words, the weight critical resonance is caused by the resonant oscillation of the weight-induced sag or bending of the rotor, at twice the shaft rotation frequency. As an example, this phenomenon can be clearly understood in connection with a board that has a rectangular cross-section and that rotates about its lengthwise axis. Such a board will have a greater weight-induced sag when its major cross-sectional dimension is horizontal, and a lesser sag when its major cross-sectional dimension is vertical (i.e. when the board is upright on edge). Thus, if the sag is measured at the longitudinal midpoint of the board as the board rotates through one revolution, the midpoint sag will be smaller when the board is at angular positions 0.degree. and 180.degree. (board upright on edge) and greater when the board is at angular positions of 90.degree. and 270.degree.. As a result, the sagging or bending of the board will cause a bending vibration having a frequency of twice the rotational frequency.
Differences or anisotropies in the flexural stiffness of a rotor can arise due to any deviation from a perfectly round cross-section of the rotor, for example due to the presence of keyways, spline grooves, or other grooves or notches. This is true even if such grooves do not effect the rotational mass balancing of the rotor. In hollow shafts or other rotors, an anisotropy can also result if any area of the inner diameter is not perfectly concentric with the outer diameter, thus causing areas of lesser and greater relative rigidity. Anisotropies in the flexural stiffness can also arise in certain regions of a cast rotor due to the presence of different grain characteristics and thus different moduli of stiffness or elasticity in these regions, as may be caused by locally different cooling velocity during the production of shafts or rolls by casting. Generally, anisotropy in the overall flexural stiffness or rigidity of the rotor can arise from any source of deviations in the flexural strength and rigidity of the rotor in one plane relative to another plane.
PCT International Publication WO 95/33143 discloses a method for reducing the flexural rigidity fluctuation and the semi-critical disturbance occurring during the operation of a roll or cylinder, whereby the measured or calculated flexural rigidity fluctuation or imbalance is compensated by forming additional grooves or pockets or by changing the size of previously machined grooves or pockets. The grooves or pockets are to be so arranged that the combined effect thereof diminishes or at least does not increase the imbalance and/or the flexural rigidity fluctuation in the roll or cylinder. This reference suggests that the required measurements can be determined by measuring the motion of the center of the roll at the midsection thereof, or by measuring the circularity profile or the diametral fluctuation after carrying out the machining operation. However, the reference does not provide any teachings regarding how and under what conditions the measurements are to be carried out, nor any teachings regarding how any measurement results are related to or used for determining compensating steps that must be carried out. Since the reference provides no direct link between measurement results and the required compensating or corrective steps, it appears that an iterative process would be required, in which repeated measurements are carried out after repeated machining steps, in a trial and error manner, to finally hopefully achieve an acceptable compensation of the flexural rigidity fluctuation.